while budget is not an issue you say you may want to specify a range. for instance $100-200 + say $100 for an amp, less than $300, etc. some headphones range in price from $5-1500+ which is why i say to give us a general idea what you think is comfortable.
the game one is based on the hd598 which is not weak in bass. it is "neutral" sounding so while bass is not prominent it has enough in relation to mids and treble. quite a decent headset if you're looking for an all in one as the hd598 is a very popular choice for people who want a neutral signature and a laid back easy listening experience.
the game zero is the bass-light model which is similar to the pc363d and pc350se its based on. not much bass at all however this gives you the most advantageous sound (can hear footsteps and similar easily).
similarly the ad500x/700x/900x are on the bass light side like the game zero but have wider soundstage.
k7xx might be okay as well as its neutral but with a bit more bass than akg normally is known for. you need good equipment and sound files for it to sound good though.. akg cans are always a bit picky.
if by "immersion" you mean fun factor and something that sounds exciting then perhaps the dt990pro-250. bright detailed treble, good low end bass for open cans, recessed mids, good soundstage, comfort. you will need a decent amp for them though.
good mics? modmic or your choice of desktop mics (atr2500usb, at2020usb, at2020+interface, blue yeti, nw800, etc)
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